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Modeling of Scheduling Batch Processor in Discrete Parts Manufacturing

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Supply Chain Strategies, Issues and Models

Abstract

Processing using batch processors (BPs) is common in many manufacturing environments. Generally in a manufacturing environment, BPs are bottleneck operations due to their longer processing time. Batch processing also involves many complexities like compatibility of jobs that can be processed, sizes of different jobs, capacity constraint of the processor, and so on. Considering all these factors, proper scheduling of BPs is important. This chapter focuses on scheduling of BPs in three real life applications, namely, automobile gear manufacturing, semiconductor manufacturing, and steel casting industries. Integer programming models are formulated for the problems under consideration. These models will help the decision makers to understand the problems better and hence work toward appropriate solutions for the problems.

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Author information

Authors and Affiliations

  1. Department of Management Studies, Indian Institute of Science, Bangalore, 560012, India

    M. Mathirajan

  2. Indian Institute of Management Indore, Indore, 453556, India

    Ravindra Gokhale

  3. School of Maritime Management, Indian Maritime University, Chennai, 600119, India

    M. Ramasubramaniam

Authors
  1. M. Mathirajan
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  2. Ravindra Gokhale
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  3. M. Ramasubramaniam
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Corresponding author

Correspondence to M. Mathirajan .

Editor information

Editors and Affiliations

  1. Department of Business Systems, University of Bedfordshire Business School, Luton, United Kingdom

    Usha Ramanathan

  2. University of Bedfordshire Business Scho, Luton, Bedfordshire, United Kingdom

    Ramakrishnan Ramanathan

Appendices

Annexure 1

A LINGO set code and sample data input for the ILP model for research problem with Hardening and Soaking Furnace (HSF)

SETS:

JOBS:RELEASE_TIME,DUE_DATE,WEIGHT,MASS,TARDINESS,CT_JOB;

FAMILIES:PROCESSING_TIME;

BATCHES:CT_BATCH,RELEASE_BATCH,PROCESSING_BATCH;

JTOF(JOBS,FAMILIES):FAM_CLASS;

FTOB(FAMILIES,BATCHES):Y;

JTOB(JOBS,BATCHES):X,FRACTION,CT_FRAC;

ENDSETS

DATA:

JOBS   =   @FILE(‘job_index.ldt');

FAMILIES   =   @FILE(‘family_index.ldt');

BATCHES   =   @FILE(‘batch_index.ldt');

RELEASE_TIME   =   @FILE(‘release.ldt');

DUE_DATE   =   @FILE(‘duedate.ldt’);

WEIGHT   =   @FILE(‘weight.ldt');

MASS   =   @FILE(‘mass.ldt');

CAPACITY   =   @FILE(‘capacity.ldt');

AVAILABILITY   =   @FILE(‘availability.ldt');

PROCESSING_TIME   =   @FILE(‘proc.ldt');

FAM_CLASS   =   @FILE(‘fam_assoc.ldt');

ENDDATA

! OBJECTIVE FUNCTION (EQUATION 1)

MIN   =   @SUM(JOBS(J): WEIGHT(J)* TARDINESS(J));

! ATMOST ONE FAMILY CAN BE PROCESSED IN A GIVEN BATCH (EQUATION 2);

@FOR(BATCHES(B):

@SUM(FAMILIES(F): Y(F,B))   <=   1);

! A BATCH CAN BE FORMED ONLY IF ITS PREVIOUS BATCH HAS BEEN FORMED (EQUATION 3);

@FOR(BATCHES(B)|B #GT# 1:

@SUM(FAMILIES(F): Y(F,B-1))   >=   @SUM(FAMILIES(F): Y(F,B)));

! RELEASE TIME, PROCESSING TIME, AND COMPLETION TIME OF BATCHES RESPECTIVELY (EQUATIONS 4, 5 AND 6);

@FOR(BATCHES(B)|B #GT# 1:

RELEASE_BATCH(B)   >=   CT_BATCH(B-1));

@FOR(BATCHES(B):

PROCESSING_BATCH(B)   >=   @SUM(FAMILIES(F):PROCESSING_TIME(F)*Y(F,B)));

@FOR(BATCHES(B):

CT_BATCH(B)   >=   RELEASE_BATCH(B)   +   PROCESSING_BATCH(B));

! A JOB CAN BE PROCESSED IN A BATCH ONLY IF THE FAMILY TO WHICH IT BELONGS TO IS PROCESSED IN THAT BATCH (EQUATION 7);

@FOR(JOBS(J):

@FOR(FAMILIES(F):

@FOR(BATCHES(B):

X(J,B) * FAM_CLASS(J,F)   <=   Y(F,B))));

! IF SOME FAMILY IS ASSIGNED TO A BATCH, THEN THAT BATCH MUST CONTAIN ATLEAST ONE JOB OF THAT FAMILY (EQUATION 8);

@FOR(FAMILIES(F):

@FOR(BATCHES(B):

@SUM(JOBS(J): X(J,B) * FAM_CLASS(J,F))   >=   Y(F,B)));

! RELEASE TIME OF A BATCH MUST NOT BE LESS THAN THE RELEASE TIME OF ANY OF JOBS CONSTITUTING THE BATCH (EQUATION 9);

@FOR(BATCHES(B):

@FOR(JOBS(J):

RELEASE_BATCH(B)   >=   RELEASE_TIME(J) * X(J,B)));

! A JOB CAN BE PROCESSED MAXIMUM IN TWO BATCHES (EQUATION 10);

@FOR(JOBS(J):

@SUM(BATCHES(B):X(J,B))   <=   2);

! A JOB MUST BE PROCESSED ATLEAST IN ONE BATCH (EQUATION 11);

@FOR(JOBS(J):

@SUM(BATCHES(B):X(J,B))   >=   1);

! THE FRACTION OF A JOB MUST BE LESS THAN OR EQUAL TO 1 (EQUATION 12);

@FOR(JOBS(J):

@FOR(BATCHES(B):FRACTION(J,B)   <=   1));

! THE FRACTION OF A JOB MUST BE GREATER THAN OR EQUAL TO 0 (EQUATION 13);

@FOR(JOBS(J):

@FOR(BATCHES(B):FRACTION(J,B)   >=   0));

! SUM OF ALL FRACTIONS FOR A JOB MUST BE 1 (EQUATION 14);

@FOR(JOBS(J):

@SUM(BATCHES(B):FRACTION(J,B))   =   1);

! IF A JOB IS ASSIGNED TO A BATCH THEN SOME FRACTION OF THAT JOB MUST BE PROCESSED IN THAT BATCH. ALSO WHEN A JOB IS NOT ASSIGNED TO A BATCH, NO FRACTION CAN BE PROCESSED (EQUATIONS 15 AND 16);

@FOR(JOBS(J):

@FOR(BATCHES(B):X(J,B)   >=   FRACTION(J,B)));

@FOR(JOBS(J):

@FOR(BATCHES(B):FRACTION(J,B)   >   10000 * (X(J,B) - 1)));

! THE QUANTITY PROCESSED IN ANY BATCH SHOULD NOT EXCEED THE BATCH PROCESSOR CAPACITY (EQUATION 17);

@FOR(BATCHES(B):

@SUM(JOBS(J): FRACTION(J,B) * MASS(J))   <=   CAPACITY);

! FRACTIONAL COMPLETION TIME OF A JOB SHOULD NOT BE LESS THAN THE COMPLETION TIME OF THE BATCH IN WHICH IT IS PROCESSED (EQUATION 18);

@FOR(JOBS(J):

@FOR(BATCHES(B):

CT_FRAC(J,B)   >=   CT_BATCH(B) - 10000 * (1 - X(J,B))));

! COMPLETION TIME OF THE JOB MUST NOT BE LESS THAN ANY OF ITS FRACTIONS (EQUATION 19);

@FOR(JOBS(J):

@FOR(BATCHES(B):CT_JOB(J)   >=   CT_FRAC(J,B)));

! RELEASE TIME OF THE FIRST BATCH ON A BP IS THE FIRST TIME AVAILABILITY OF THE BP (EQUATION 20);

@FOR(BATCHES(B)|B #EQ# 1:RELEASE_BATCH(B)   >=   AVAILABILITY);

! DEFINE TARDINESS (EQUATIONS 21 AND 22);

@FOR(JOBS(J): TARDINESS(J)   >=   CT_JOB(J) - DUE_DATE(J));

@FOR(JOBS(J): TARDINESS(J)   >=   0);

! ASSIGNS BINARY RESTRICTION ON THE DECISION VARIABLES (EQUATION 23 AND 24)

@FOR(JTOB:@BIN(X));

@FOR(FTOB:@BIN(Y));

The sample input files for LINGO implementation, for the numerical example described in Sect. 2.4 (Table 3) are given below

Input file ‘job_index.ldt’: A B C D E F G H I J

Input file ‘family_index.ldt’: P Q R

Input file ‘batch_index.ldt’: a b c d e f g h i j

Input file ‘release.ldt’:

Input file ‘duedate.ldt’:

Input file ‘weight.ldt’:

Input file ‘mass.ldt’:

Input file ‘fam_assoc.ldt’:

6

36

6

20

1 0 0

1

15

4

18

0 1 0

10

52

6

13

0 1 0

10

22

5

19

0 0 1

4

14

3

3

1 0 0

0

20

3

11

1 0 0

0

10

5

11

1 0 0

2

32

6

2

1 0 0

7

27

3

12

1 0 0

6

24

2

6

0 0 1

  1. Input file ‘capacity.ldt’: 30 Input file ‘availability.ldt’: 1
  2. Input file ‘proc.ldt’:
  3. 10
  4. 14
  5. 6

Annexure 2

Problem instance wise optimal total weighted tardiness (TWT) and required computational time for research problem with hardening and soaking furnace (HSF)

Problem instance

Optimal TWT

Time required to obtain optimal TWT (hr-min-sec)

Problem instance

Optimal TWT

Time required to obtain optimal TWT (hr-min-sec)

Problem instance

Optimal TWT

Time required to obtain optimal TWT (hr-min-sec)

1

250

00-00-32

33

83

00-01-12

65

432

00-26-57

2

0

00-00-01

34

0

00-00-02

66

30

00-00-16

3

93

00-00-05

35

112

00-00-10

67

200

00-03-55

4

230

00-00-37

36

1

00-00-01

68

205

00-01-44

5

190

00-00-11

37

97

00-00-03

69

122

00-00-21

6

30

00-00-01

38

96

00-00-05

70

324

00-18-00

7

40

00-00-11

39

51

00-00-02

71

254

00-11-25

8

36

00-00-04

40

32

00-00-03

72

111

00-00-27

9

208

00-00-32

41

307

00-02-01

73

347

00-17-33

10

133

00-00-07

42

146

00-00-38

74

27

00-00-08

11

106

00-00-10

43

0

00-00-01

75

389

00-08-50

12

24

00-00-06

44

64

00-00-12

76

74

00-03-41

13

268

00-02-53

45

28

00-00-10

77

212

00-02-40

14

95

00-00-03

46

6

00-00-06

78

66

00-00-32

15

187

00-00-24

47

69

00-00-14

79

347

00-09-39

16

48

00-00-12

48

3

00-00-02

80

0

00-00-01

17

369

00-01-51

49

73

00-00-04

81

67

00-01-04

18

0

00-00-01

50

14

00-00-14

82

131

00-00-19

19

81

00-00-07

51

64

00-00-08

83

152

00-01-48

20

4

00-00-04

52

43

00-00-11

84

29

00-00-51

21

121

00-00-27

53

220

00-00-23

85

54

00-00-18

22

34

00-10-03

54

210

00-00-21

86

280

00-02-17

23

20

00-00-01

55

60

00-00-01

87

106

00-01-12

24

68

00-00-02

56

0

00-00-01

88

85

00-00-29

25

250

00-00-27

57

61

00-00-11

89

151

00-00-50

26

41

00-00-01

58

44

00-00-07

90

68

00-00-32

27

156

00-00-07

59

220

00-01-06

91

111

00-03-47

28

14

00-00-02

60

48

00-00-02

92

0

00-00-01

29

185

00-00-20

61

205

00-00-21

93

257

00-03-57

30

76

00-00-02

62

24

00-00-11

94

90

00-00-11

31

141

00-00-20

63

137

00-00-19

95

167

00-00-52

32

0

00-00-03

64

48

00-00-05

96

0

00-00-14

97

277

00-25-40

129

667

02-22-04

161

373

02-32-33

98

62

00-02-10

130

35

00-01-39

162

147

00-12-46

99

217

00-08-49

131

275

00-56-47

163

426

03-26-48

100

0

00-00-01

132

124

00-07-35

164

109

00-16-37

101

147

00-04-19

133

495

12-29-19

165

574

02-50-40

102

101

00-04-30

134

178

00-30-27

166

12

00-01-02

103

184

00-08-28

135

196

01-02-27

167

215

01-20-32

104

117

00-01-42

136

35

00-03-26

168

78

00-04-00

105

84

00-00-01

137

452

03-51-31

169

528

02-38-17

106

164

00-03-40

138

164

00-04-53

170

99

00-07-12

107

98

00-02-36

139

301

01-04-06

171

277

02-29-49

108

0

00-00-03

140

13

00-00-43

172

56

00-09-33

109

434

01-11-33

141

302

02-53-37

173

179

00-36-19

110

33

00-00-09

142

284

00-07-00

174

17

00-02-30

111

149

00-06-06

143

124

00-05-23

175

220

00-29-22

112

226

00-05-33

144

176

00-27-51

176

48

00-07-27

113

339

00-24-55

145

204

00-11-32

177

169

00-14-08

114

115

00-01-55

146

86

00-04-10

178

76

00-10-41

115

201

00-08-43

147

150

00-14-40

179

200

00-27-29

116

3

00-00-45

148

58

00-01-41

180

119

00-38-35

117

223

00-11-16

149

487

01-59-33

181

310

01-03-00

118

257

00-03-16

150

82

00-00-35

182

208

00-35-32

119

312

00-18-05

151

61

00-01-02

183

198

01-25-14

120

24

00-00-17

152

91

00-05-11

184

102

00-12-48

121

401

00-08-52

153

263

02-26-40

185

279

00-45-40

122

60

00-00-18

154

288

00-15-02

186

288

00-32-20

123

18

00-00-07

155

297

00-23-08

187

149

00-11-18

124

208

00-02-30

156

0

00-00-29

188

161

00-33-31

125

229

00-03-59

157

199

00-06-16

189

279

01-16-51

126

325

00-18-28

158

0

00-00-16

190

20

00-03-04

127

122

00-01-50

159

201

00-27-07

191

93

00-08-38

128

101

00-00-50

160

54

00-14-03

192

0

00-00-30

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Mathirajan, M., Gokhale, R., Ramasubramaniam, M. (2014). Modeling of Scheduling Batch Processor in Discrete Parts Manufacturing. In: Ramanathan, U., Ramanathan, R. (eds) Supply Chain Strategies, Issues and Models. Springer, London. https://doi.org/10.1007/978-1-4471-5352-8_7

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